How to see the eight Thurston geometries

05/26/2020 ∙ by Tiago Novello, et al. ∙ IMPA 0

A manifold is a topological space that is locally Euclidean. Manifolds are important because they arise naturally in a variety of mathematical and physical applications as global objects with simpler local structure. In this paper we propose a technique for immersive visualization of relevant three-dimensional manifolds in the context of the Geometrization conjecture. The algorithm generalizes traditional computer graphics ray tracing. To do so we use several related definitions and results dating back to the works of Poincaré, Thurston, and Perelman.

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1 Introduction

A -manifold is a space locally similar to the -dimensional Euclidean space, but in which the global structure may be non-trivial. Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed in terms of the relatively well-understood properties of simpler spaces.

In this paper we investigate the problem of visualizing -manifolds, which is not as easy as visualizing the Euclidean space. Specifically, we set our scene objects at the -manifold spaces. Inspired on [3], we avoid modeling perspective views by generalizing the ray tracing algorithm: a color is given to each point and tangent direction by tracing a ray and finding its intersections with the scene objects. Recall from physics that light travels along with rays: paths that locally minimize lengths. Tracing a ray requires geometry; finding a “nice” one is a hard task as we will see along this text.

We think of -manifolds as spaces representing the shape of the universe since, from our eyes, they look like the Euclidean space. This is a three-dimensional version of the fact, for example, that the earth (a -sphere) is locally similar to a plane. For a -manifold example, consider the set of points equidistant from a fixed point in the four-dimensional Euclidean space — the -sphere. This space plays a central role in the study of -manifolds being the main actor in Poincaré conjecture.

The dimension is a hard constraint on -manifolds viewing; our eyes only see up to dimension three. -Manifolds can be visualized extrinsically using a three-dimensional Euclidean space to illustrate its universal covering, and intrinsically by embedding the oriented surface in the Euclidean space and visualizing it through classical algorithms: rasterization or ray tracing.

The problem of visualizing -manifolds is harder. However, in 1998, Thurston published How to see -manifolds  [34], discussing ways to visualize a -manifold using our spatial imagination and computer aid. Many tools in -manifold theory are inspired in human spatial geometrical instincts. Thus, the human mind is trained to understand the kinds of geometry that are needed for 3-manifolds. Finding a “geometry” for a given -manifold is related to the Thurston’s geometrization conjecture, which encapsulates Poincaré conjecture [17]. The conjecture states that each -manifold admits a unique geometric structure that can be associated with it. This paper presents and visualizes these geometric structures.

Since higher dimensional manifolds can not be used to visualize -manifolds, we take an immersive approach based on a ray tracing algorithm. Rasterization is not appropriate for this scenario because perspective projection in non-Euclidean spaces is computationally nontrivial. In the other hand, a scene embedded in a -manifold can be ray traced: given a point (eye) and a direction (from the eye to the pixel), we trace a geodesic (ray). When it reaches an object we compute its shading.

2 History

2.1 Henri Poincaré

In 1895, Henri Poincaré published his Analysis situs [26], in which he presented the foundations of topology by proposing to study spaces under continuous deformations; position is not important. The main tools for topology are introduced in this paper: manifolds, homeomorphisms, homology, and the fundamental group. He also discussed about how three-dimensional geometry was real and interesting. However, there was a confusion in his paper: Poincaré treated homology and homotopy as equivalent concepts.

In 1904, Poincaré wrote the fifth supplement [27] to Analysis situs, where he approached -dimensional manifolds. This paper clarified that homology was not equivalent to homotopy in dimension three. He presented the Poincaré dodecahedron as an example of a -manifold with trivial homology but with nontrivial homotopy. In Subsection 6.1, we present an inside view of such space. Poincaré proposed the conjecture: Is the -sphere the unique compact connected -manifold with trivial homotopy?

Poincaré stimulated a lot of mathematical works asking whether some -manifold exists. Works on this question were awarded three Fields medals. In 1960, Stephen Smale proved [30] the conjecture for -manifolds with . In 1980, Michael Freedman proved [8] Poincaré conjecture for -manifolds. The problem in dimension three was open until 2003 when Grisha Perelman proved [23, 25, 24] Thurston’s geometrization conjecture, and consequently the Poincaré conjecture as a corollary.

Poincaré also worked on an important problem in dimension two, the uniformization theorem. This states that every simply connected Riemann surface (one-dimensional complex manifolds) is conformally equivalent to the unit disc, the complex plane, or the Riemann sphere. This was conjectured by Poincaré in 1882 and Klein in 1883, and proved by Poincaré and Koebe in 1907. The history details can be found in the recent book by Ghys [9]. A big step in the history of the geometry was the generalization of this result for dimension three, Thurston’s geometrization theorem.

2.2 William P. Thurston

Thurston’s works in -manifolds have a geometric taste with roots in topology. He tried to generalize the uniformization theorem of compact surfaces to dimension three. Five more geometries arise; hyperbolic geometries still playing the central role.

In 1982, Thurston stated the geometrization conjecture [35] with solid justifications. It is a three-dimensional version of the uniformization theorem, where hyperbolic geometry is the most abundant because it models all surfaces with genus greater than one. In dimension three, Thurston [35] proved that the conjecture holds for a huge class of -manifolds, the Haken manifold, implying that hyperbolic plays, again, the central role. The result is known as the hyperbolization theorem. Thurston received in 1982 a Fields medal for his contributions to -manifolds. The elliptzation conjecture, the part which deals with spherical manifolds, was open at that time.

2.3 Grisha Perelman

In 2000, the Clay Institute selected seven problems in mathematics to guide mathematicians in their research, the seven Millennium Prize Problems [13]. Poincaré conjecture was one of them. The institute offered one million dollars for the first proof of each problem. They did not know that the Poincaré conjecture was about to be solved by Grisha Perelman as a corollary of the proof of the geometrization conjecture.

In 2003, Perelman published three papers [23, 25, 24], in arXiv solving the Geometrization conjecture. He used tools from geometry and analysis. Specifically, he used the Ricci flow, a technique introduced by Richard Hamilton to prove the Poincaré conjecture. Hamilton proved the conjecture for a special case when the -manifold has positive Ricci curvature. The idea is to use Ricci flow to simplify the geometry along time. However, this procedure may create singularities since this flow expands regions with negative Ricci curvature and contracts regions of positive Ricci curvature. Hamilton suggested the use of surgery before the manifold collapse. The procedure gives rise to a simpler manifold, and we can evolve the flow again. Perelman, proved that this algorithm stops and each connected component of the resulting manifold admits one of the Thurston geometries. In other words, Perelman proved the geometrization conjecture, and consequently the Poincaré conjecture. Seven research groups around the world have verified his proof.

3 Basic Concepts

Several main concepts are needed to relate -manifolds and ray tracing. We start with some definitions on topology of manifolds, then we associate a geometry to them.

3.1 Topology

Topology is the branch of mathematics that studies the shape of objects modulo continuous deformation. Informally, we can stretch, twist, crumple, and bend, but not tear or paste. -Manifolds are examples of topological spaces that are locally similar to the -dimensional Euclidean space. Loops are examples of -manifolds, and compact surfaces are examples of -manifolds, including the sphere, and the torus.

To understand a manifold it is common to use its fundamental group. This records basic information about the shape (holes) of the space. Introduced by Poincaré, the fundamental group consists of equivalent classes under continuous deformation of loops contained in the space. A manifold is simple connected if its fundamental group is trivial. Poincaré conjecture states that each compact simply connected -manifold must have the -sphere shape. This implied in the discovery of many manifold constructions.

A common manifold construction is through the quotient of “simpler” manifolds by special groups acting on them. This is reasonable because each manifold is uniquely covered by a simple connected manifold: the universal covering [16]. Informally, a manifold covers a manifold if there is a map which “evenly covers” a neighborhood of each point in . The covering is universal if is simple connected. For example, the torus is covered by Euclidean space. The Poincaré conjecture implies that if the universal covering of a compact manifold is compact, then it must be the sphere. By the above discussion, we only need to consider quotients of simply connected manifolds.

Let be a manifold and be a discrete group acting on it. The quotient manifold theorem (Theorem 9.16 in [16]) states that is a manifold when the group acts smoothly and properly discontinuous on . The action is properly discontinuous if each point admits a neighborhood such that , for all different from the identity. If is a compact surface, it is the torus or the Klein bottle [17].

More examples of manifolds can be constructed from the direct product. For example, the -torus is the product of the circle and the -torus .

3.2 Geometry

In Riemannian geometry, manifolds receive a metric which allows the introduction of geodesics: paths that locally minimize lengths. These are the ingredients for a ray tracing algorithm on manifolds. Following the notation of Carmo [4], we present a brief introduction of the definitions and examples of Riemannian geometry.

Every point in a -manifold admits a neighborhood homeomorphic to the open ball of , the correspondent maps are called charts. We need the change of charts in to be differentiable. Let be a chart of a neighborhood of a point . The tangent space at

is the vector space spanned by the tangent vectors

of the coordinate curves at . A Riemannian metric in is a map assigning a scalar product to each tangent space, such that in coordinates, , the function is smooth. Expressing two vectors in terms of the associated basis, that is, and , we obtain:

(1)

The metric is determined by . The pair is a Riemannian manifold. For examples, consider the classical geometries: Euclidean, hyperbolic, and spherical spaces, as well as the non-classic: , , Nil, Sol, and (see Section 6).

Let and be Riemannian manifolds of dimension and , the -manifold admits a Riemannian metric given by , the product metric. Examples include the geometries and .

Lie groups are important examples of Riamannian manifolds. A Lie group is a group where its operations and are smooth. Thus its left multiplication by , given by , is smooth. The classical way to define a Riemannian metric in a Lie group is by fixing a scalar product in the tangent space at the identity element , and extend it by left multiplication:

(2)

Nil, Sol, and geometries are examples of Lie groups, see Subsection 6.3.

Quotient of Riemannian manifolds by discrete groups produces new manifolds. Specifically, the quotient of a Riemannian manifold , by a discrete group acting isometrically on it, has the geometric structure modeled by . This quotient is equal to a covering, so we consider being simply connected. There are exactly three Riemannian surfaces modeling the geometry of all closed compact surfaces (see Section 4.1). In dimension three the list is increased by five especial examples of product, and Lie group geometries. These are model geometries: complete simply connected Riemannian manifolds such that each pair of points have isometric neighborhoods.

We define the main ingredient of the ray tracing. A geodesic in a Riemannian manifold is a curve with null covariant derivative:

(3)

This differs from the classical by the addition of , which includes the Christoffel symbols of . To linearize System 3, we add new variables being the first derivatives , obtaining thus the geodesic flow of :

(4)

4 Two-dimensional manifolds

We present some well-known results involving topology and geometry of surfaces. We assume all surfaces been compact, connected, and oriented. Starting with the classification theorem in terms of the connected sum, one can represent a surface through a polygon with an appropriate edge gluing. This polygon can be embedded in one of the three two-dimensional geometry models (Euclidean, spherical, and hyperbolic). The resulting surface has the geometry modeled by one of these geometries.

4.1 Classification of compact surfaces

The classical way to state the classification theorem of surfaces is by the connected sum. Removing disks and from surfaces and , one obtains their connect sum by identifying the boundaries and through a homeomorphism. The theorem says that any compact surface is homeomorphic to a sphere or a connected sum of tori.

The proof of the classification theorem uses a computational representation of a compact surface through an appropriate pair-wise identification of edges in a polygon:

  • Take a triangulation of ; it is a well-known result;

  • Cutting along edges in we obtain a list of triangles embedded in the plane without intersection; the edge pairing must be remembered;

  • We label each triangle edge with a letter according to its gluing orientation;

  • Gluing the triangles through its pairwise edge identification without leaving the plane produces a polygon . The boundary is an oriented sequence of letters;

  • Let and be a couple of edges in . If the identification of and reverses the orientation of we denote by , and simply otherwise;

  • A technical result states that by cutting and gluing leads us to an equivalent polygon with its boundary having one of following configurations:

    • , which is a sphere;

    • , a connected sum of tori .

To model the geometry of those surfaces, we embed, in a special way, the polygon in one of the two-dimensional model geometries.

4.2 Geometrization of compact surfaces

We remind the well-known geometrization theorem of compact surfaces which states that any topological surface can be modeled using only three geometries.

Theorem 4.1 (Geometrization of surfaces).

Any compact surface admits a geometric structure modeled by the Euclidean, the hyperbolic, or the spherical space.

The Euclidean space models the geometry of the -torus through the quotient of by the group of translations. The sphere is modeled by the spherical geometry.

For a hyperbolic surface, consider the bitorus, which topologically is the connect sum of two tori. The bitorus is presented as a regular polygon with sides as discussed above. All vertices in are identified into a unique vertex . Then, the corners of are glued together producing a topological disk. Considering with the Euclidean geometry, the angular sum around equals to . To avoid such a problem, let be a regular polygon centered in the hyperbolic plane, with an appropriate scale, its angles sum . The edge pairing of induces a group action in the hyperbolic plane such that is the bitorus. In terms of tessellation, tessellates with regular -gons. Analogously, all surfaces represented as polygons with more than four sides are hyperbolic. Implying that hyperbolic is the most abundant geometry.

The above discussion handled all orientable surfaces. The well-known Gauss–Bonnet theorem implies that these geometric structures must be unique.

5 Three-dimensional manifolds

It took time to formulate the modern idea of a manifold in a higher dimension. For example, a version of Theorem 4.1 for -manifold seemed not possible until 1982, when Thurston proposed the geometrization conjecture [35]. It states that each -manifold decomposes into pieces shaped by simple geometries. There are eight geometries in dimension 3, which are presented in more detail in Section 6. Scott [29] is a great text on this subject.

5.1 Classification of compact 3-manifolds

As for surfaces, there is a combinatorial procedure to build three-dimensional manifolds from identifications of polyhedral faces.

To do so, endow a finite number of polyhedra with an appropriate pair-wise identification of its faces. Each couple of faces has the same number of edges and it is mapped homeomorphically to each other. Such gluing gives a polyhedral complex , which is a -manifold iff its Euler characteristic is equal to zero (Theorem 4.3 in [7]).

We now take the opposite approach. Let be a compact -manifold, we represent as a polytope endowed with a pair-wise identification of its faces. The following algorithm mimics the surface case presented in Subsection 4.1.

  • Let be a triangulation of ; endorsed by the well-known triangulation theorem;

  • Detaching every face identification in gives rise to a collection of tetrahedra which can be embedded in . Remember the pairwise face gluing;

  • Gluing in a topological way each possible coupled tetrahedra without leaving produces a polytope . The faces in the boundary are pairwise identified.

The combinatorial problem of reducing to a standard form, as in the surface case, remains open (see page 145 in [16]). Although there is not (yet) a classification of compact -manifold in the sense presented for compact surfaces, it is still possible to decompose the given manifold in simpler pieces. This decomposition is not trivial. Thurston conjectured that these pieces can be modeled by eight geometries.

The decomposition used in the geometrization theorem (to be presented in Section 5.2) has two stages: the prime and the tori decomposition. The first is similar to the inverse of the connected sum of surfaces. It consists of cutting the -manifold along a -sphere such that the resulting two disconnected -manifolds are not balls. Attaching balls in the boundary of these parts one obtain a simpler -manifold. A prime -manifold does not admit such sphere decomposition. Kneser proved that, after a finite number of steps, a manifold factorizes into prime manifolds, and Milnor proved that the decomposition is unique [19] up to homeomorphism.

Tori decomposition [14, 12] consists of cutting a prime -manifold along “certain” tori embedded. The result is a -manifold bounded by tori that are left as boundaries, because there is no canonical way to close such holes.

Decomposing a -manifold through the above procedure produces a list of simpler manifolds, which resembles an evolutionary tree [18]. Each of these manifolds is modeled by one of eight (Thurston’s) geometries. This is the -dimensional case of Theorem 4.1: the Thurston–Perelman geometrization theorem. See Figure 1.

5.2 Geometrization of compact 3-manifolds

The geometrization of surfaces is controlled by the Euler characteristic. -manifolds are more complicated. Thurston [35] proposed that the simpler manifolds given by the prime and torus decomposition admit the geometric structure of eight geometries, the geometrization conjecture. It is not always possible to give a single geometry to the manifold. These geometries include Euclidean, hyperbolic, and spherical spaces.

Theorem 5.1 (Geometrization).

Any compact, topological -manifold can be constructed using just geometry models.

The other five geometries are the product spaces and , endowed with the product metric, and the -dimensional Lie groups Nil, Sol, and . All the eight geometries are homogeneous, that is, for every pair of points, there is a local isometry sending one to another. Only Euclidean, hyperbolic, and spherical spaces have isotropic geometries, that is, isometries on the tangent space at every point can be realized as isometries of the underlying manifold. We present an informal description and more details of Thurston geometries in Section 6.

We explain the word construct in Theorem 5.1. A -manifold is geometrically modeled by one of Thurston geometries if it is the quotient of such spaces by a discrete group. Prime and tori decomposition provides the candidate -manifolds to be modeled by Thurston geometries (the leaves in Figure 1). The geometrization theorem factorizes the manifold into pieces modeled by the eight geometries.

In the surface geometrization, hyperbolic geometry played a central role. The same happens in dimension three, most of the eight geometries are required to describe particular manifolds. Thurston said [35] that hyperbolic geometry is by far the most interesting, the most complex, and the most useful among the eight geometries. The other seven play only in exceptional cases. In Section 6 we present some ideas explaining the abundance of manifolds modeled by hyperbolic geometries.

The geometrization theorem implies the Poincaré conjecture. A compact simply connected -manifold is a prime manifold, also it does not contain a torus non-trivially embedded (since its fundamental group is trivial). The geometrization theorem implies that the manifold is modeled by one of the eight geometries. As the fundamental group is finite, the manifold must be the quotient of the sphere by a discrete group (Elliptization conjecture), which should be trivial since it is isomorphic to the fundamental group.

At this point, we should clarify two hard questions. Why are there exactly eight geometries? How can Theorem 5.1 be proved? We present some informal intuitions and ideas of the proofs. The first question is approached in Section 6. The technique using Gauss-Bonnet theorem does not work in this case.

Perelman’s proof of the geometrization theorem involves geometry and analysis tools that are beyond the scope of this paper. Very informally, Perelman’s argument consists of starting from a -manifold endowed with a Riemannian metric . Then running Hamilton’s Ricci flow , where is the metric which evolves along time controlled by the Ricci curvature. This evolution smooths the metric giving a more “uniform” shape to the manifold (similar to the heat equation). This procedure may produce singularities since (in some sense) the differential equation may create critical elements. Perelman overcomes this by cutting the manifold into certain pieces (prime and tori decomposition) just before the collapse appears. Then he repeats the method on each of the individual pieces. He proved that this algorithm decomposes the manifold in a “tree” with each leaf been a manifold with geometry modeled by one of the Thurston geometries, see Figure 1.

Figure 1: Evolutionary tree of a compact orientable -manifold. It operates like an algorithm. The first two layers indicate the prime and tori decomposition of the -manifold. The last two is the geometrization theorem.

6 The eight Thurston Geometries

We presented the geometrization of -manifolds: each manifold decomposes into pieces shaped by eight homogeneous geometries. Here we provide the definitions and some features of these geometries. We justify why the hyperbolic geometry is the richest, presenting all the manifolds modeled by the Thurston geometries. For a rigorous presentation of the eight Thurston geometries, see [29, 35, 17].

The classification mentioned above uses the concept of Seifert manifolds: closed manifolds admitting a decomposition in terms of disjoint circles. Martelli [17] describes two results. The first states that a closed orientable -manifold can be geometrically modeled by one of the following six geometries: iff it belongs to a special class of Seifert manifolds. It has a Sol geometric structure iff it admits a particular torus bundle, called torus semi-bundle of Anosov type.

The second result states that if a -manifold admits a geometric structure modeled by one of Thurston geometries, it is specified by the manifold fundamental group:

Fundamental group Model geometry
Finite
Virtually cyclic
Virtually abelian
Virtually nilpotent
Virtually solvable
Contains a normal cyclic group
Quotient lifts
a finite-index subgroup
Otherwise
Otherwise

We skip these group definitions because they deviate from the scope of this paper. The hyperbolic abundance is due to restriction of the seven group classes aforementioned.

Thurston geometries can be divided in three classes. The isotropic geometries (Euclidean, spherical, and hyperbolic spaces) are called classical. The product geometries are and . , , and are the Lie group geometries. All these geometries are homogeneous, every pair of points admits similar neighborhoods. The classical geometries admit constant sectional curvature since they are isotropic [4].

6.1 Classical

For dimension exists a unique complete, simply connected Riemannian manifold having constant sectional curvature , , or . These are the sphere, the Euclidean space, and the hyperbolic space. Conversely, if a complete manifold has constant sectional curvature , , or , it must be the quotient of such models geometries by a discrete group (Proposition 4.3 in [4]). We present these geometries, examples of manifolds modeled by them, and the behavior of rays in such spaces.

Euclidean space

In dimension two, every orientation preserving isometry in Euclidean space is a translation. Then, if is a compact orientable surface, it must be the torus (see Section 6.2 of Martelli [17]). In dimension three this list is increased by five more orientable manifolds since we can compose rotations with translations.

The Euclidean space is with the inner product , where and are vectors in . The distance between two points and is . The curve describes a ray leaving a point in a direction . Analogously, for any the Euclidean space is constructed.

For an example of a -manifold modeled by , consider the flat torus , obtained by gluing opposite faces of the unit cube in . is also the quotient of by its group of translation spanned by , , and . The unit cube is the fundamental domain.

A ray leaving a point in a direction is parameterized as . For each intersection between and a face of the unit cube, we update by in the opposite face; is normal to . The direction does not need to be updated.

Then, we have the ingredients for an inside view of . The fundamental domain receives the scene and the rays in can return to it, resulting in many copies of the scene. The immersive perception is tessellated by scene copies; see Figure 2.

Figure 2: Inside view in the flat torus. We use the cube to set up our scene: a unique mesh (Suzzane) with hands and the cube’s edges. The face pairing makes the rays that leave a face return from its opposite.

Beyond the torus, there are exactly five more compact oriented -manifold with geometry modeled by the Euclidean geometry, see Figure 3.

Figure 3: The six compact oriented flat manifolds. These are built through pair-wise gluing: faces are identified isometrically according to their labels, otherwise, it is glued to its opposite in an obvious way. From [17].

Hyperbolic space


Hyperbolic space can be described in many ways, unlike Euclidean and spherical spaces. Here we present the hyperboloid and Klein models and a manifold modeled by such rich space. There are plenty of hyperbolic manifolds, making this concept a central actor in the topology of -manifolds [17].

The Lorentzian space is with the product , where . The hyperbolic space is the hyperboloid endowed with the metric , where and are points in . Due to its similarity to the sphere definition, is known as pseudo-sphere.

A tangent vector to a point in satisfies . Moreover, the tangent space coincides with the set . The Lorentzian inner product is positive on each tangent space.

Rays in arise from intersections between and planes in containing the origin. A ray leaving a point in a tangent direction is the intersection between and the plane spanned by the vectors and . Its parameterization is . Thus, rays in can not be straight lines.

It is possible to model in the unit open ball in — known as the Klein model — such that in this model the rays are straight lines. More precisely, each point is projected in the space by considering , the space is obtained by forgetting the coordinate .

The hyperbolic space is a model of a Non-Euclidean geometry, since it does not satisfy the Parallel Postulate: given a ray and a point , there is a unique ray parallel to . For a ray in the hyperbolic space and a point there is an infinite number of rays going through which do not intersect .

For a compact -manifold modeled by hyperbolic geometry considers the Seifert–Weber dodecahedral space. It is the dodecahedron with an identification of its opposite faces with a clockwise rotation of . The face pairing groups edges into six groups of five, making it impossible to use Euclidean geometry. The regular Euclidean dodecahedron has a dihedral angle of degrees. The desired dodecahedron should have a dihedral angle of degrees, which is possible in hyperbolic space considering an appropriate dodecahedron diameter.

Then, we ray trace Seifert–Weber dodecahedron. A ray leaving a point in a direction is given by since we are using Klein’s model. For each intersection between and a dodecahedron face, we update and through the hyperbolic isometry that produces the face pairing above. This isometry is quite distinct from Euclidean isometries [11]. The immersive perception of using this approach is a tessellation of by dodecahedra with a dihedral angle of degrees. Figure 4 illustrates an inside view of the Seifert–Weber dodecahedral space.

Figure 4: Immersive view of Seifert–Weber dodecahedron. We use the dodecahedron to set up our scene: a unique Suzzane with hands and the dodecahedron’s edges. The face pairing make the rays that leave a face return, with an additional rotation, from its opposite face, giving rise to many copies of the scene: a tessellation of the hyperbolic space by rotated dodecahedra.

Spherical space


The -sphere is with the metric . As in the hyperbolic case, a tangent vector to a point in satisfies . The tangent space corresponds to the set . The space inherits the Euclidean inner product of .

A ray in passing through a point in a tangent direction is the arc produced by intersecting with the plane spanned by , , and the origin of . Such ray is parameterized as .

is a Non-Euclidean geometry because it fails the Parallel Postulate: given a ray and a point , there is a unique ray parallel to . As the rays in are the big arcs, intersecting two of then in always result in exactly two intersecting points.

Gluing the opposite faces of the dodecahedron with a clockwise rotation of we get the Poincaré dodecahedral space (or Poincaré homological sphere); its first homological group is trivial. The face pairing groups edges into ten groups of three edges. Then, we need a dodecahedron with dihedral angle of . In this case, we use spherical geometry by finding a dodecahedron in the -sphere with an appropriate diameter. The immersive visualization of Poincaré dodecahedral space is a tessellation of by dodecahedra. This is a -dimensional regular polytope: the -cell (see Figure 5).

Figure 5: Inside view in the Poincaré dodecahedron. The Suzzane with hands and the dodecahedron’s edges composes teh scene. The faces pairing make the rays iterate tessellating the sphere, the -cell.

6.2 Product geometry

The eight three dimensional geometries include products of lower-dimensional geometries, which are and endowed with the product metric. We do not present (yet) immersive visualization of them because they model few manifolds [17].

Space 
The geometry models very few manifolds. The sectional curvature is along with horizontal directions and along with verticals. Recall that sectional curvature of a plane is the Gauss curvature associated with the surface generated by such a plane.

Consider the manifold endowed with the product metric for an example of a manifold modeled by . The geometry of can not be modeled by classical geometries, since has as it universal covering and it is not isotropic.

Space 
The geometry is given by the product metric. Analogous to the geometry horizontal and vertical planes have sectional curvature and .

6.3 Lie group geometry

The remaining three non-isotropic geometries to analyze are not products, but they admit a kind of “bundle structure”. The first attempt to visualize these geometries in real-time (using VR) appeared in 2019 [1]. See [5, 6, 28, 15] for other great works.

Nil space


Nil geometry is a -bundle over . This geometry is constructed from the Lie group called the Heisenberg group [17].  [20] describes in more detail.

Nil space () is an example of a Lie group consisting of all real matrices

with the multiplication operation. There is a natural identification of with .

The multiplication of elements in is the sum of its coordinates, with an additional term in the last one. This term makes all the difference, since in order to put a geometry in we consider the left multiplication , for all , being isometries.

We construct a metric in Nil by considering the Euclidean product in the tangent space at . Then we extend it by left multiplication. After some calculations we obtain the scalar product between the tangent vectors and at a point

The matrix above defines a metric at . Varying we obtain a Riemannian metric , since each matrix entry is differentiable. The vectors , , and form an orthogonal basis at . Also, the volume form of coincides with the standard one from , since the metric determinant is unity.

The geodesic flow on admits a solution [32]. A ray starting at in the tangent direction is given by:

To compute a geodesic starting at in the direction , we translate the initial conditions to the origin, then using the solution above we compute the geodesic. We translate this back to the desired position.

For a compact manifold modeled by consider the discrete group generated by the “translations” in the axis direction , , and . inherits the geometry of . For each fixed we obtain a two-dimensional torus; is foliated by tori. The unit cube is the fundamental domain. Figure 6 gives an inside view.

Figure 6: Inside view in a Nil manifold. A hand and the cube’s edges compose the scene. The face pairing makes the rays iterate giving a tessellation of by cubes.

space


The geometry is similar to Nil, but it is now a -bundle over . The geometry is constructed from the Lie group , in a certain way.  [21] describes in more detail. Here we present only its main features.

We follow the notation of Gilmore [10]. The special linear group consisting of all matrices with unit determinant is a Lie group: the product of two matrices with unit determinant has unit determinant, the same for the inverse matrix.

To understand the hyperbolic nature of observe that the elements of are matrices such that . Then is a -manifold embedded in given by . Rewriting , we get:

which describes the equation of a -hyperbola in .

There is also an identification of with ; see [21] for more details. That is, is not simply connected, which implies that it is not a model geometry. However, the universal cover of is a model geometry [33]. We focus on the visualization of since their geometries are locally identical.

We use the parameterization of a neighborhood of identity [10]:

(5)

Observe that is the identity of , and that in the plane the map is not defined. We use to push-back the metric of to .

Then, we construct a metric in the . The element is the identity of . Let be the tangent space at with the well-known scalar product between two tangent vectors and  [10]. As in Nil geometry, we extend it to a Riemannian metric using left multiplication.

Using the above Riemannian metric we obtain the geodesic flow (see [21]):

(6)

which can be solved using Euler’s method. Figure 7 gives a visualization of .

Figure 7: Immersive view in a local parameterization of . The scene is a grid in deformed by the metric.

Sol space


Sol is the least symmetric among the eight geometries. It is a plane bundle over the real line. Its geometry comes from a Lie group Sol. For details see [22].

The Sol space () is an example of a Lie group which consists of all matrices

with the multiplication operation. Clearly, is diffeomorphic to .

Let and be elements in . Their multiplication has the form:

which is the sum of the element coordinates controlled by an additional term in the first coordinates. To endow with a geometry we consider the Euclidean metric in the tangent space at the origin and extend it by left multiplication. After some computations we get the scalar product of two tangent vectors and at :

The matrix above defines a metric at . Varying we obtain a Riemannian metric , since each matrix entry is differentiable. The volume form of coincides with the standard one from , since the determinant of the above matrix is one.

Using the above metric we obtain the geodesic flow of the Sol geometry:

(7)

Sadly, there is no solution to this problem in terms of elementary functions [31]. Troyanov [36] obtained a formula for geodesics in

, however, it contains many coefficients that can not be computed in a closed formula. He classified

geodesics in classes of equivalence, the horizons of .

For a compact manifold modeled by the Sol space, consider to be the discrete group generated by the “translations” in the direction of axis , , and . inherits the geometry of Sol space and for each fixed it provides a two-dimensional torus, thus admits a foliation by torus. The unit cube is the fundamental domain. Figure 8 presents the visualization of this manifold.

Figure 8: Inside view in a Sol manifold. The cube’s edges give the scene. The face pairing makes the rays that leave a face return from its opposite, tessellating .

7 Beyond the Canonical Spaces

Until now we presented the definitions and results that allows us to endow complex topological spaces with simpler geometries. However, another direction can be taken: given a geometry, we want to deform it. Many applications dating back to the work of Barr [2] arises from this approach. Additionally, we are interested into investigating Riemannian geometry to find new approaches for shading. We present two examples: manifolds as graphs of functions and manifolds as deformations of .

7.1 Graph of a function

The graph of a smooth function is the three dimensional manifold:

(8)

The structure of has a unique chart . The tangent space in a point is generated by ; is the partial derivative of in the standard direction . The Euclidean metric of induces a metric in .

Let be a point, and , be tangent vectors of . Expressing and in terms of the tangent space basis, and . Applying Equation 1 we obtain the metric: .

As are smooth functions in , is a Riemannian metric on . The pair is a Riemannian manifold. Note that , thus the volume form of only coincides with the standard one of when .

Computing the geodesic flow of through Equation 4, we model an immersive visualization in this manifold, see Figure 9.

Figure 9: Inside view of a graph of saddle function. The scene is a regular grid in . The Riemannian metric deforms the grid.

7.2 Diffeomorphisms

Let , given by , be a smooth map which admits a smooth inverse, a diffeomorphism. The deformation provides a metric to . The base manifold is and the parameterization is . The associated base of is

(9)

We pull-back the Euclidean metric of through the differential of .

Let be a point, and , be tangent vectors at . Expressing and in terms of the basis given by Equation 9, then and . Applying Equations 1 and 9 we obtain the metric:

(10)

As is smooth, is a Riemannian metric. The pair is a Riemannian manifold.

Computing the geodesic flow of through Equation 4, we model an immersive visualization in this manifold (see Figure 10).

Figure 10: Inside view of a twisted . The scene is a regular grid in .

We generalize the concept of shading for Riemannian geometry. This would allow us to create a more general model to compute ray tracing.

7.3 Riemannian shading and illumination

Shading is the process of assigning a color to a pixel and illumination is the attribution of a color to a surface point by simulating light attributes. Sometimes these terms are used interchangeably. We define shading and illumination in the context of Riemannian geometry. The idea is to visualize scenes embedded in -manifolds.

For shading, we consider a -sphere centered in a point (the eye). This sphere carries the image. Then we give a color for each sphere point (ray direction) by tracing a ray. We call this procedure Riemannian shading. Specifically, the unit sphere is centered at the origin of . For each direction in we attribute a color by launching a ray from in the direction . Each time intersects a scene object at a point we define an RGB color based on the object properties. Therefore, we obtain the Riemannian shading , where is a color space.

Riemannian illumination is the process of defining a color for a point in a surface , given a light source and an eye . The transport of light from the sources to the point is done by the direct geodesic (ray) connecting to or indirect geodesics. The computation of the direct rays is a very hard problem. The indirect ones are easier because we could use a path tracer: the rays can be integrated using the geodesic flow of the manifold. The local illumination of almost coincides with the classical, only the inner product must be changed.

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